SPLINE-TYPE SPACES IN GABOR ANALYSIS

H. G. FEICHTINGERDepartment of Mathematics, University Vienna

Strudlhofg. 4, A-1090 Vienna, AustriaE-mail: hans.feichtinger@univie.ac.at

The central topic in Gabor analysis are Weyl-Heisenberg families, obtained from a“window” g, by means of time-frequency shifts along some lattice Λ � IR2d. As astarting point we consider tight Gabor frames, i.e., we choose g ∈ L2(IRd) in sucha way that the corresponding frame operator is represents the identity:

f =∑λ∈Λ

〈f, π(λ)g〉π(λ)g ∀f ∈ L2(IRd).

Using such expansions one can define Gabor multipliers via

Gmf =∑λ∈Λ

m(λ)〈f, π(λ)g〉π(λ)g.

The main results of this paper will answer the question under which conditionsthe mapping from the upper symbol (m(λ)) to the Gabor multiplier Gm (for given(g, Λ)) is injective on `∞, and how the best approximation of a given operator(e.g., some Hilbert-Schmidt operator) by Gabor multipliers (with `2-symbol) canbe described. In order to prove these results we shall make use of propertiesof spline-type spaces over locally compact Abelian groups, generated by meansof atoms in the modulation space M1,1

0 (IRd), also known as Feichtinger’s Segalalgebra S0(G). The corresponding results make an original part of this paper.

This article will appear essentially in its present format as acontribution to "Wavelet Analysis: Twenty Years’ Developments",(D. X. Zhou, ed.) World Scientific Press, Singapore, 2002. Thecore material of this contribution has been presented at theBommerholz Conference on Multivariate Approximation in September2000, organized by K.Jetter et al..

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1 Introduction

It is the purpose of this paper to present some basic facts concerning Gabormultipliers. These are operators obtained by multiplying the (canonical) Gaborcoefficients with a real- or complex-valued sequence defined over the time-frequency lattice in use. In order to ensure that the constant multiplier m(λ) =1 yields the identity operator, and that real-valued multiplier sequences yieldself-adjoint operators, it is convenient to assume in many cases that the Gaborexpansion in use is obtained from a tight Gabor atom (in this case the dualGabor atom coincides with the Gabor atom itself).

In order to ensure suitable continuity properties of Gabor operators it isrelevant to recall the definition and some basic properties of the Segal algebraS0(IRd) = M1,1

0 (IRd), which is also a member of the family of modulation spaces(cf. [Gro01,RS01]). Gabor atoms from this space have much better propertiesthan general L2-atoms. Moreover, there are many sufficient conditions onf ∈ L2 to ensure that f ∈ S0(IRd) (see [FZ98,Gro01]). However, this spaceis also relevant for a discussion of spline-type spaces on IRd. These spacesare spanned by a Riesz basis, which is obtained from some atom by takingtranslates along some discrete subgroup of IRd. For the case of splines of orderr ∈ N the generating atom is simple the B-spline of order r.

Section 3 discusses Riesz basis which are obtained as translates of someatom (e.g. B-spline) along some lattice. It is well known that the canonicalbiorthogonal Riesz basis is again a collection of translates of some dual atom.In the general L2-theory this dual atom belongs to the spline-type space, i.e.the closed linear span of the original Riesz basis. Equivalently, it can be repre-sented as an infinite series of shifted atoms with `2-coefficients. Using Wiener’sinversion theorem we show that dual atom can even be represented with `1-coefficients, if S0-atoms are used. As as consequence the projection operatorP from L2(IRd) onto the spline-type space also continuous with respect to Lp-norms, for the full range of p ∈ [1,∞]. Its range are the functions which canbe expanded using `p-coefficients. Furthermore P is also even to a boundedlinear mapping on S′0(IR

d) and S0(IRd), respectively.Section 4 then collects various basic facts about Weyl-Heisenberg frames

or Riesz bases obtained by time-frequency shifts of some Gabor atom. Againatoms from the space S0(IRd) play an important role.

Section 5 represents the core of the present paper. After describing the ba-sic properties of Gabor multiplier operators we discuss conditions which ensurethat the mapping from the multiplier sequence in `2(Λ) into the Hilbert spaceHS of Hilbert-Schmidt operators is injective. It turns out that the relevantproperty is a Riesz-basis property of the family of rank ones operators, project-

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ing onto the elements of the given Gabor frame. Once this Riesz basis propertyis characterized by means of a condition on the sampled Wigner function ofg, it is possible to describe the best approximation of general Hilbert-Schmidtoperators can be described, using the canonical biorthogonal basis.

The connection to spline-type spaces established via the Kohn-Nirenbergcorrespondence is then used to show that these properties extend to larger(resp. smaller) spaces of operators. Thus for example injectivity of the mappingfrom multiplier sequences to Gabor multipliers is also injective on `∞(Λ).

Throughout this paper the concept of Gelfand triples (as already system-atically used in [FK98]) helps to compactly express the continuity propertiesof the various linear mappings relevant for this paper.

2 Notations and Basic Facts

In this section we briefly recall the basic concepts of Fourier Analysis and theconcept of an STFT (Short-time Fourier transform). For simplicity, we workin the setting of Euclidean spaces (G = IRd), although most of the resultsextend without major changes to the setting of locally compact Abelian (=LCA) groups. (See[Fol89],[RS01] for general background).

First, recall that the Lp-spaces over IRd, 1 ≤ p ≤ ∞, are taken withrespect to the natural Lebesgue measure, the space L2(IRd) taking a specialrole as Hilbert space. These spaces are isometrically invariant under translationoperators Tx, x ∈ IRd, defined by Txf(z) = f(z − x), and the modulationoperators Ms, given as Msf(z) = e2πis·z, for s ∈ IRd, i.e., modulation is justmultiplication of a function by the character χs(t) := e2πis·t, where we interprets·t as the scalar product between the vectors t, s ∈ IRd. When combining themwe prefer to think of time-frequency shift operators π(λ) = MsTt for λ = (t, s)

being a general point in the time-frequency planea IRd × IRd.

It is an important feature of TF-analysis that time frequency shifts do notcommute, since Ms1Tx1Ms2Tx2 = e−2πix1s2Ms1+s2Tx1+x2 . Therefore, chang-ing the order in an arbitrary product of TF-shifts introduces phase factors ofabsolute value one. For this reason the mapping λ = (t, s) 7→ π(λ) is only a

(faithful = injective) projective representation of IRd × IRd.

The Short-time Fourier transform of an L2-function f with respect tosome window function g ∈ L2(IRd) is defined as a function over the TF-planeby STFTg(f)(t, s) = 〈f,MsTtg〉. For fixed t it is—as a function of s—the

aThe .-symbol in the second variable indicates that it is understood as frequency variable;furthermore operators are applied in the following order: first time shift, then frequencyshift, whereas the symbols show up in reverse order.

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Fourier transform of f · Ttg, describing the local frequency content of f near t(and usually displayed in the y-direction).

Since we are going to work with the Fourier transform normalized as

f(s) =∫

IRd

f(x)e−2πix·sdx ,

Plancherel’s theorem states that F : f 7→ f is a unitary mapping, and itsinverse is obtained by replacing the exponential term in the definition of theFT by its conjugate. This normalization of the Fourier transform also impliesthat for each lattice Λ � IRn, of the form Λ = A(ZZd), for some non-singularn× n-matrix A, the orthogonal subgroup Λ⊥ turns out to be Λ⊥ = A∗−1(ZZd),with A∗ = conj(At).

As a consequence of Plancherel’s theorem the Short-time Fourier transformsatisfies

‖STFTg(f)‖2 = ‖f‖2 · ‖g‖2 for f, g ∈ L2(IRd). (2.1)

Hence for ‖g‖2 = 1 the mapping f 7→ STFTg(f) is an isometric map fromL2(IRd) into L2(IR2d), so that its adjoint mapping inverts the STFT on itsrange. This gives the standard reconstruction formula for the STFT for nor-malized g:

f =∫

IRd×IRd

STFTg(f)(t, s) MsTt g dtds. (2.2)

Furthermore it is essentially a consequence of the Cauchy-Schwarz inequalitythat STFTgf is indeed a bounded and continuous function on IRd × IR

d, with

‖STFTgf‖∞ ≤ ‖f‖2‖g‖2.Whenever g is not just in L2, but for example in the Schwartz-space of

rapidly decreasing functions, one can take the STFT for any tempered distri-bution and finds that it is a continuous function over IRd × IR

dwith at most

polynomial growth. It is therefore possible and indeed very useful to definea family of function spaces (usually called modulation spaces over IRd). Theyare characterized by the (global) behaviour of the STFT of their elements (cf.[Gro01] for a detailed discussion). Indeed, according to the theory of coorbitspaces [FG-I,FG-II], for any weighted Lp-space the space of tempered distri-butions with STFTg(f) in that space is a well-defined Banach space withrespect to its natural norm, and different non-zero “windows” g define thesame space with equivalent norms. Therefore it is convenient to choose g = g0,with g0(t) = e−πt2 the Fourier-invariant Gauss function, as the window in thedefinition of such spaces.

Besides L2(IRd) (which corresponds to L2(IRd × IRd) under the correspon-

dence principle alluded in the previous paragraph), the modulation spaces

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which correspond to L1(IR2d) and L∞(IR2d) are most relevant for our discus-sions below. In the literature the first one is often denoted by S0(IRd), alsoknown as Feichtinger’s algebra ( see [Gro01,RS01]), and is given as

S0(IRd) = {f ∈ L2(IRd), STFTg0(f) ∈ L1(IRd × IRd)}, (2.3)

with natural norm ‖f‖S0 := ‖STFTg0f‖L1 . The corresponding space using anL∞-condition turns out to be the Banach dual of S0(IRd):

S′0(IRd) = {f ∈ S ′(IRd), STFTg0(f) ∈ L∞(IRd × IR

d)},

with the norm ‖f‖S′0

:= sup(t,s)|STFTg0f(t, s)|.Without proof we also note that the Sobolev space with exponent p =

1 and derivatives of order two is embedded into S0(IR). In other words, iff, f ′, f ′′ ∈ L1(IR), then f ∈ S0(IR). Various characterizations of S0(IRd) aregiven in [FG92], for example.

In the sequel we need a couple of basic results concerning these two spaces(cf. [FZ98] and[FK98]). Various characterizations of S0(IRd) are given in [Z98],or Sec. 12.1 of [Gro01].

Theorem 2.1. (Basic Properties of S0(IRd))(1) (S0(IRd), ‖·‖S0

) is a Banach space, which is continuously embedded into(Lp(IRd), ‖·‖p), for 1 ≤ p ≤ ∞. TF-shifts act isometrically on S0(IRd), i.e.,

‖π(λ)(f)‖S0 = ‖f‖S0 ∀λ ∈ IRd × IRd, ∀f ∈ S0(IRd);

(2) S0(IRd) is invariant under the Fourier transform as well as under arbitraryautomorphisms of IRd (such as dilations);(3) For any lattice Λ � IRd there exists a constant C1

Λ such that∑λ∈Λ

|f(λ)| ≤ C1Λ‖f‖S0 ∀f ∈ S0(IRd).

As a consequence S0(IRd) is a natural domain for the Poisson summationformula (PSF): There exists CΛ > 0 such that∑

λ∈Λ

f(λ) = CΛ

∑λ⊥∈Λ⊥

f(λ⊥) ∀f ∈ S0(IRd). (2.4)

(4) For f ∈ S0(IRd) and for any lattice Λ, the Λ-periodization of f (obtainedby the absolutely convergent sum fΛ(x) :=

∑λ∈Λ f(x + λ)) has an absolutely

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convergent Fourier series expansion, i.e., it belongs to the Wiener’s algebraA(IRd/Λ) of absolutely convergent Fourier series.

Proof. The first three statements are covered by the references indicatedabove. The last statement is essentially a combination of the first two ones,since the Fourier coefficients of fΛ are exactly the sampling values of f (whichbelongs to S0(IRd) for f ∈ S0(IRd)) over the orthogonal lattice Λ⊥.

The isometric invariance of (S0(IRd), ‖·‖S0) under the Fourier transform as

well as arbitrary TF-shifts can be used (via transposition) to obtain the sameproperties for its dual (S′0(IR

d), ‖·‖S′0). Since it is furthermore possible to show

that complex interpolation (with the interpolation parameter θ = 1/2) appliedto the pair (S0(IRd), S′0(IR

d)) yields just the Hilbert space L2(IRd), it is oftenvery useful to consider these spaces not just in isolation, but as a so-calledGelfand triple (cf. [FK98] for details on this concept). Clearly S0(IRd) is adense subspace of the Hilbert space L2(IRd), and in consequence L2(IRd) isdense in S′0(IR

d) in the w∗-sense. We shall simply write (S0, L2, S′0) if we

talk about this particular Gelfand triple. A unitary isomorphism of Gelfandtriples has to be understood as a linear mapping which maps one Gelfandtriple into another one in the following sense: It is a unitary mapping at theHilbert space level, continuous (and hence an isomorphism) at the level of theunderlying smaller Banach spaces S0(IRd), and extends to their dual spaces(being there not only norm continuous, but also w∗-w∗-continuous).

In this particular case a way of understanding the connections betweenthese spaces is via the so-called Wilson bases (cf. again [FK98], or section 12.3in [Gro01]). A function f ∈ L2(IRd) belongs to S0(IRd) if and only it has`1-coefficients with respect to a typical Wilson basis, whereas the elementsof S′0(IR

d) are exactly those (tempered) distributions which have a Wilsonexpansion with `∞-coefficients.

A typical situation where this concept provides interesting extra informa-tion is the so-called kernel theorem. It is one of the many instances whereS0(IRd) and its dual show properties very similar to the Schwartz space andits dual, the space of all tempered distributions (see [Fei80], [FG92] or [Gro01Section 14.4] for details, as usual we write (f ⊗ g)(x, s) = f(x)g(s)):

Theorem 2.2. (Kernel Theorem)(1) There is a natural isomorphism T 7→ κ(T ) between the Banach spaceL(S0, S

′0) of bounded linear operators T from S0(IRd) to S′0(IR

d) and theirdistributional kernels in S′0(IR

2d), satisfying κ(T )(f ⊗ g) = Tf(g).(2) If κ(T ) ∈ S0(IR2d) then T extends to a unique bounded linear mappingfrom S′0(IR

d) to S0(IRd) with the property of mapping w∗-convergent sequencesin S′0(IR

d) into norm convergent sequences in S0(IRd).(3) If the kernel κ(T ) belongs to L2(IR2d) then T extends to a compact operator

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on L2(IRd); in fact, the operators arising in this way are exactly the Hilbert-Schmidt operators, and the mapping κ(T ) 7→ T is a unitary mapping betweenL2(IR2d) and HS b.

Proof. A detailed description of these identifications, using the terminologyof Gelfand triples, is given in [FK98].

One can view the above identification of operators with their (distribu-tional) kernels as a unitary Gelfand triple isomorphism between the Gelfandtriple (S0, L

2, S′0)(IR2d) and the triple (B,HS,B′), where B is defined as the

space of linear operators on L2(IRd) with S0-kernel. Operators in B are traceclass on L2 and map S′0(IR

d) into S0(IRd); B′ can be identified with L(S0, S′0)

(as Banach spaces with their respective natural norms).Finally, we will need a particular unitary representation ρ of a given

TF-lattice Λ (any discrete subgroup of IR2d with compact quotient) on theHilbert-Schmidt operators over L2(IRd). We proceed as follows. First we con-sider the mapping λ 7→ π(λ), which is only a projective representation on theHilbert space L2(IRd). However, it can be used to define ρ on the Hilbertspace HS of all Hilbert Schmidt operators over G = IRd by defining the actionof λ ∈ Λ on a given operator T by conjugation of T by the operator π(λ).Therefore one usually writes π ⊗ π∗ for the representation ρ described in thefollowing lemma (cf. [FK98] for details):

Lemma 2.3. The mapping λ 7→ ρ(λ), given by ρ(λ)T := π(λ)◦T ◦π(λ)−1,

defines a unitary representation of IRd × IRd

on the Hilbert space HS of allHilbert-Schmidt operators on L2(IRd) which leaves B and B′ invariant.

3 Spline-type spaces over LCA groups

In this chapter we want to state various results concerning spline-type spacesover general locally compact Abelian = LCA groups. They are meant to benon-trivial (although reminiscent of folklore results) even for the Euclideangroup IRd. We assume for simplicity that the generating elements are in theSegal algebra S0(IRd), as this is the situation which will be relevant for ourapplications to Gabor analysis.c In fact, we are going more for the generalityin terms of the groups for which such statements can be made (i.e., arbitraryLCA groups), than for the minimal conditions on the atoms.

Let us recall the definition of spline-type space (sometimes also called prin-cipal shift-invariant spaces, as they are generated by a single function and its

bThe natural scalar product for HS is 〈T1, T2〉 = trace(T1 ◦ T ∗2 ).cUniform p-stability can also be shown under weaker assumptions, such as membership ofthe atom in the Wiener algebra W (C0, `1).

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translates) over general LCA groups. Before doing so we remind the readerthat a Banach space of sequences Bd is called solid if for any sequence c ∈ Bd

and d with |d| ≤ |c| (in the pointwise sense), d ∈ Bd and ‖d‖Bd≤ ‖c‖Bd

.

Definition 3.4. Let Λ be any discrete subgroup of G and g be any el-ement in some translation-invariant Banach space (B, ‖·‖B) of functions ordistributions on G. Then the closed linear span of the family (Tλg)λ∈Λ is aΛ-invariant subspace of (B, ‖·‖B). We denote this space by Vg,Λ, and call it aspline-type space (in B) if the family (Tλg)λ∈Λ is a Riesz projection basisfor Vg,Λ, i.e., if the following two properties hold:d

(1) There is some naturally solid Banach space (Bd, ‖·‖Bd) of sequences on Λ

(naturally associated with (B, ‖·‖B)) such that the synthesis mapping (repre-sentation operator R) described by R : c 7→

∑λ∈Λ cλTλg is well-defined and

defines a continuous bijection between Bd and Vg,Λ.(2) There exists a bounded linear mapping C (coefficient mapping) defined onall of B such that C ◦R = IdVg,Λ .

Remarks: The existence of the left inverse C to the given synthesis op-erator R is in fact equivalent to the assumption that there exists a boundedand linear projection operator P from B onto Vg,Λ. Indeed, if the conditionsof our definition are satisfied, then the mapping P = R ◦C is such a bounded(and idempotent) linear mapping whose range is exactly Vg,Λ. Conversely, ifcondition (1) is satisfied and we can find such a projection, then C = R−1 ◦ Pis a well-defined left inverse to R (the inverse being understood as being theinverse on the range of R, which is Vg,Λ by assumption). Thus the definitionimplies that a spline-type space in B is a closed, complemented subspace of B.If the finite sequences are dense in Bd then the family (Tλg) is an unconditionalbasis (as the image of the unit vectors in Bd under R) for Vg,Λ.

The concept of Riesz projection bases has been introduced byG.Zimmermann in his thesis ([GZDiss]). Details on this concept, already in-dicating its usefulness in the framework of Gabor analysis, can be found inChap. 3 of [FS98]. Clearly, for the case that (B, ‖·‖B) is a Hilbert space suchas L2(G) (and the associated solid Banach space is `2(Λ)), it is enough tocare for assumption (1), as for any closed subspace in a Hilbert space one canuse the orthogonal projection, which is of norm 1. Thus in the case of Hilbertspaces, condition (1) can be rephrased in the usual way by assuming that there

dOur use of the terminology is more restrictive than that by most other authors, includingAldroubi/Grochenig in [AG01], However, in terms of relevant examples we do not excludeany of the practically useful ones, but rather emphasize some extra properties shared byBanach spaces of cubic splines on IR.

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exist constants C,D > 0 such that for all c ∈ `2(Λ) one has:

C‖c‖2 ≤ ‖∑λ∈Λ

cλTλg‖ ≤ D‖c‖2.

Indeed, such an estimate follows from the fact that Vg,Λ is a closed subspace of(B, ‖·‖B) using Banach’s theorem. Conversely, it is evident that the validityof this norm equivalence not only implies that the range of R is exactly Vg,Λ,but also that the family (Tλg)λ∈Λ is a Riesz basis for Vg,Λ, and therefore has acanonical biorthogonal system in Vg,Λ (cf. [You80] for general characterizationsof Riesz bases in Hilbert spaces).

We summarize our findings by stating that such a family is a Riesz basisfor a Vg,Λ if and only if it is an `2(Λ)-Riesz projection basis. It is also an`1(Λ)-Riesz projection basis for Vg,Λ ∩ L1(IRd) (with the L1-norm). Indeed,this is equivalent to the fact that not only g but also g belong to L1(IRd),where g is the generator for the biorthogonal Riesz basis (Tλg)λ∈Λ. Followingstandard terminology we call g the dual atom to g (with respect to Λ)e.

Consequently, the subspace Vg,Λ ∩ L1(IRd) coincides exactly with thoseelements in Vg,Λ which have `1-coefficients in their standard expansion. Moregenerally it follows that one has `p-stability for arbitrary values of p ∈ [1,∞] inthis case, i.e., the Lp-norm of a function of the form

∑λ∈Λ cλTλg is equivalent

to the `p-norm of the sequence (cλ)λ∈Λ (cf. also [AST01]).In other words, in this situation there are constants C1, C2 (not depending

on p ∈ [1,∞]), such that for any such p one has norm equivalence between the`p-norm of the coefficients, and the continuous Lp-norm of the function f ∈Vg,Λ. Of course, one can extend such statements easily to weighted Lp-spaces(for example to weights of polynomial growth up to a certain order) whenever gsatisfies stronger (decay) conditions. These are certainly satisfied for Schwartzfunctions g.

It will be also clear from the comments given so far that it will often occurthat the mappings R and C occurring in the context of Riesz projection basiscan often be understood not just as mappings for an individual pair of Banachspace (B, ‖·‖B) an (Bd, ‖·‖Bd

), but actually between families of compatibleBanach spaces in the sense of interpolation theory.

Just for later reference (and to illustrate the generality of our definitionmodeled after the Hilbert space case) let us formulate a simple general resultfirst. The proof is left to the interested reader.

Lemma 3.5. Assume that Vg,Λ is a spline-type space within (B, ‖·‖B),such that the “finite” sequences are dense in (Bd, ‖·‖Bd

). Then also V ′g,Λ

eStrictly speaking one should indicate a possible dependence on the norm of B.

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is a spline-type space (with respect to B′d). The “biorthogonal” (via the

B-B′-duality) system to (Tλg) is of the form (Tλg), where g is called the dualatom.

For our applications to Gabor multipliers the following results will be veryuseful, showing that the dual atom g will be “of the same quality as” g. In away it is a consequence of the Wiener-Levy theorem (cf. [RS01]).

Theorem 3.6. (Spline-type spaces with S0(G)-atom)Let g ∈ S0(G) be given, and let Λ � G be a lattice.(1) The family (Tλg)λ∈Λ is a Riesz basis (for its closed linear span Vg,Λ) if andonly if the Λ⊥-periodized version of g, i.e., H :=

∑λ⊥∈Λ⊥ |Tλ⊥(g)|2, is free of

zeros.(2) In this case,

(a) the Fourier transform of the function g generating the biorthogonal Rieszbasis (Tλg)λ∈Λ is given by g = g/H, and g belongs to S0(G);

(b) there is also some g2 ∈ S0(G) such that (Tλg2)λ∈Λ is an orthonormalbasis for Vg,Λ, with g2 = g/

√H; and

(c) if in addition g ≥ 0, then the Lagrange interpolation problem over Λ:f(0) = 1, f(λ) = 0 for 0 6= λ ∈ Λ, has a unique solution in gL ∈ Vg,Λ,characterized by its Fourier transform gL = g/

∑λ⊥∈Λ⊥(Tλ⊥ g). Furthermore

gL ∈ S0(G).Corollary 3.7. For (g,Λ) as above, the families (TλgL)λ∈Λ and (Tλg)λ∈Λ

induce a Riesz projection basis for all spaces V pg,Λ, defined as the closed linear

span of (Tλg)λ∈Λ in Lp, for 1 ≤ p < ∞, establishing a natural isomorphismbetween these spaces and the corresponding `p-spaces over Λ.

Proof of Thm.3.6: The characterization of the Riesz basis property of afamily (Tλg)λ∈Λ (within L2(G)) through the condition

H(s) :=∑

λ⊥∈Λ⊥

|g(s− λ⊥)|2 ≥ δ0 > 0 ∀s ∈ G (3.5)

is well-known for many different cases and works for general groups just inthe same way as for IRd (see [OCL89] for an early reference, but also [Jia97],[BL98], or [AG01] for a recent survey article). It is not difficult to recognizethat condition (3.5) is equivalent to the bounded invertibility of the Gram-matrix as operator on `2(Λ), whose entries are 〈Tλ1g, Tλ2g〉. More explicitly,its entries are the sampling values of the autocorrelation function g ∗ g∗ for gover Λ. Evidently it is a circulant matrix with respect to the group structureof the Abelian group Λ. This already implies that in case of invertibility theinverse Gramian matrix is again circulant, and hence the biorthogonal familyconsists of shifted versions of some dual atom, which we will call g. Moreover,

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on the Fourier transform side g is characterized as g = g/H. Next we have toshow that g ∈ S0(IRd) as well.

First we look closer at the convolution kernel representing the actionGramian matrix on `2(Λ). By a standard argument its Fourier transformcoincides with the Λ⊥ periodized version of φ = |g|2. Since g ∈ S0(G) impliesφ ∈ S0(G) (by Thm. 2.1(2)), this periodized version H is certainly a continuousfunction. Evidently it is free of zeros if (and only if) its inverse is a boundedfunction. Let us assume now that this is the case. Using Thm. 2.1(4) we findthat H also belongs to the Wiener amalgam space H ∈ W (A, `∞), resp. to thealgebra of absolutely convergent Fourier series over G/Λ⊥. Hence the Wiener-Levy inversion theorem (see [RS01], Chap.1.3 and Thm.6.1.1) can be applied,and its inverse has the same property. In other words 1/H is an absolutelyconvergent sum of Λ⊥-periodic characters, or

1H(s)

=∑λ∈Λ

r−λχλ for all s ∈ G, with r ∈ `1(Λ).

By applying the inverse Fourier transform we find that g =∑

λ∈Λ rλTλg.Since S0(G) is isometrically translation invariant we have g ∈ S0(G) (with‖g‖S0 ≤ ‖r‖1‖g‖S0).

The rest of Theorem 3.6 will not be used and its proof (in analogy to corre-sponding results in the literature) is left to the reader. Once the usual formulafor the Lagrange interpolator is established only a mild modification of theabove argument is needed in order to find out that this Lagrange interpolatorwithin the spline type space belongs to S0(G) as well.

Corollary 3.8. (i) For (g,Λ) as above the orthogonal projection fromL2(IRd) onto Vg,Λ can be described alternatively as

f 7→ Pg,Λ(f) =∑λ∈Λ

[f ∗ g∗](λ) · Tλg =∑λ∈Λ

[f ∗ g∗](λ) · Tλg. (3.6)

This mapping is not only bounded with respect to the L2-norm, but also (uni-formly) bounded with respect to all Lp-norms, with p ∈ [1,∞].

(ii) If furthermore g(s) ≥ 0 for all s ∈ G we also have

Pg,Λ(f) =∑λ∈Λ

f(λ) · TλgL,

with the same continuity properties.In this situation a function f ∈ Lp can be written as Lp convergent series of

translates of g if and only if its sampling sequence (f(λ))λ∈Λ belongs to `p(Λ).

11

Indeed, the mapping between the coefficients and the sampling values can bedescribed as a convolution operator (over Λ) with `1(Λ)-convolution kernel forthe operator as well as its inverse.

Proof. One checks that 〈f, Tλg〉 = f ∗ g∗(λ), where g∗(z) = g(−z). Forfurther details see also [Kan95].

Remark: As a simple consequence of the above results one has for (g,Λ) asabove the following for 1 ≤ p < ∞: If (Tλg)λ∈Λ is a Riesz basis in L2(IRd), thenf ∈ Lp(IRd) belongs to the closed linear span (in the Lp-norm) of the translatesof g if and only if it has a representation with `p-coefficients. Furthermore, theLp-norm and the W (C, `p)-norms are equivalent on Vg,Λ. For the case p = 1the L1 and the W (C, `1)-norm are furthermore equivalent to the S0-norm onthe space Vg,Λ ∩ L1.

Remark: Of course, the reconstruction formula given in (ii) above re-sembles very much the classical Shannon sampling formula, where one hasg = SINC, the cardinal sine, which has the interesting property of formingboth an orthogonal family within its closed linear span (Vg,Λ is just a Paley-Wiener space of band-limited functions) and being the Lagrange interpolatorfor this space. However, due to its non-integrability this form of reconstructionis limited to band-limited functions in Lp, for the range p ∈ (1,∞), and thecorresponding bounds are deteriorating towards the critical values p = 1 andp = ∞, respectively.

4 Basic facts about Weyl-Heisenberg families

The key object of Gabor Analysis are the so-called Weyl-Heisenberg families,which are obtained from some given function g (also called Gabor atom, typi-cally a Schwartz function, such as some Gaussian “window”) by time-frequencyshifts along some lattice Λ. This lattice is a discrete subgroup of G× G withcompact quotient, such as aZZd × bZZd

� IRd × IRd, for a pair (a, b) of positive

lattice constants. The latter case is usually called the separable case. Moregenerally, non-separable subgroups are obtained by applying a suitable invert-ible 2d × 2d-matrix A to the standard lattice ZZ2d. Throughout this paper,we will not make use of the concrete form of A, but describe the situation ingeneral group-theoretical terms. This allows the statements to be interpretedas results valid for general locally compact Abelian groups G instead of justfor IRd.

At least if g is a “decent function”, and if the lattice constants are suffi-ciently small (i.e., the corresponding lattices Λ are accordingly dense in G×G),it is not too difficult to verify that the corresponding family (π(λ)g)λ∈Λ formsa frame for L2(G), i.e., we have the following two-sided estimate, for some

12

positive numbers A,B > 0 and all f ∈ L2(G) (cf. [FZ98]):

A‖f‖2 ≤∑λ∈Λ

|〈f, π(λ)g〉|2 ≤ B‖f‖2.

Since the middle term 〈f, π(λ)g〉 coincides with STFTg(f)(λ), this condi-tion amounts to the assumption that there is norm equivalence between theL2-norm and the `2(Λ)-norm of the Short-time Fourier transform of f with re-spect to the window g, sampled at Λ. Indeed, if the lattice is rather dense thistwo-sided estimate can be explained by saying that up to normalization (herethe size of A and B comes in) the middle term is essentially a Riemannian sumfor the integral of |STFTg(f)|2 (cf. Eq. (2.1)).

The so-called Gabor frame operator is defined for f ∈ L2(G) by

S(f) = Sg,Λ(f) :=∑λ∈Λ

〈f, π(λ)g〉π(λ)g.

S is not only bounded on L2(G) and strictly positive, hence invertible, butalso commutes with all TF-shifts from Λ, i.e., it satisfies π(λ) ◦ S = S ◦ π(λ)for all λ ∈ Λ. This implies that the dual frame is again a Weyl-Heisenbergfamily with respect to the same TF-lattice Λ, generated by the dual Gaboratom g. In other words, the simple identity S ◦ S−1 = Id = S−1 ◦ S impliesthat one has the double coherent frame expansionf valid for any f ∈ L2(G)(with unconditional convergence in L2):

f =∑λ∈Λ

〈f, π(λ)g〉π(λ)g =∑λ∈Λ

〈f, π(λ)g〉π(λ)g.

Here we have used that g = S−1g generates the dual frame, i.e., the family{S−1(gλ)}, using exactly the same TF-shifts used to generate the originalGabor frame, since

S−1(π(λ)g) = π(λ)(S−1g).

Convention: We write from now on briefly gλ for π(λ)g and gλ for π(λ)g,without danger of confusion. The Gabor frame representation formula forf ∈ L2(G) thus reads, emphasizing the double coherence in analogy to splineexpansions,

f =∑λ∈Λ

〈f, gλ〉gλ =∑λ∈Λ

〈f, gλ〉gλ.

fWe use the term of double coherence if the same family of operators is applied to just twodual atoms to generate a pair of dual frames.

13

Indeed, such an expansion should also be seen similar to the so-calledφ-transform expansions as developed by Frazier-Jawerth [FJ1,FJ2].

For the case that A = B we speak of a tight Gabor frame, which impliesthat Sg,Λ = A · Id and consequently g = g/A. In other words the frameexpansion looks very much like an orthonormal expansion, i.e.,

f =1A

∑λ∈Λ

〈f, gλ〉gλ,

with the only difference that its building blocks are not linear independentin the following sense: any of its elements can be expressed as (absolutely)convergent series involving all the other atoms.

In both cases it is very plausible that a good concentration of g and g (andtheir Fourier transforms) around the origin implies that two time-frequencyshifted copies (of g resp. g) of the form π(λ1)g and π(λ2)g are almost orthogonalto each other if |λ1 − λ2| is large. Therefore it is not so surprising that Weyl-Heisenberg families (π(λ◦)g)λ◦∈Λ◦ with respect to any sufficiently coarse latticeΛ◦ form a Riesz basis for their closed subspace (which we denote by V = Vg,Λ◦)in L2(G), i.e., there are positive constants C,D > 0 (depending only on Λ◦)such that for any sequence c ∈ `2(Λ)

C‖c‖2 ≤ ‖∑

λ◦∈Λ◦

cλ◦gλ◦‖2 ≤ D‖c‖2.

Again it is not difficult to verify that the canonical biorthogonal family, whichis also contained in V , is uniquely determined, and can also be obtained froma single Gabor atom by TF-shifts along Λ◦.

Membership of g in S0(G) is also a sufficient condition for the boundedness(“Bessel property”) of the relevant operators in Gabor analysis.

Proposition 4.9. For g ∈ S0(G), Λ � G× G discrete, one has:

f 7→ Sg(f)|Λ = (〈f, π(λ)g〉)λ∈Λ

and its adjoint, which is of the form

c 7→∑λ∈Λ

cλπ(λ)g,

are continuous linear mappings between the following Banach spaces:(i) S0(G) and `1(Λ),(ii) L2(G) and `2(Λ), and(iii) S′0(G) and `∞(Λ).

14

If (π(λ)g)λ∈Λ is a Gabor frame, then the dual Gabor window can be usedto determine the canonical, i.e., the least squares coefficients, via

cλ(f) = 〈f, π(λ)g〉 = Sgf(λ), ∀λ ∈ Λ. (4.7)

If both g and g are sufficiently well-behaved functions generating a pairof dual Gabor frames (cf. [FZ98] for details) one derives therefrom a charac-terization of function spaces (more precisely, a variety of modulation spaces)through the behaviour of their sampled STFT over Λ. More precisely one has:

Proposition 4.10. Let g ∈ S0(IRd) be given such that (g,Λ) gener-ates a Gabor frame, and assume that g ∈ S0(IRd) as well. Then one has:

f ∈ S0(G) ⇔ (Sgf(λ))λ∈Λ ∈ `1(Λ),f ∈ L2(G) ⇔ (Sgf(λ))λ∈Λ ∈ `2(Λ),f ∈ S′0(G) ⇔ (Sgf(λ))λ∈Λ ∈ `∞(Λ).

Remark: It is a remarkable fact, shown in Grochenig-Leinert ([GL01],the rational case was treated in [FG97]), that it is in fact not necessary toassume that g ∈ S0(IRd), as this follows automatically under the first part ofthe assumption. Indeed, if (gλ)λ∈Λ is a frame in the Hilbert space H = L2(IRd)generated by an element in S0(IRd), then it is also true that g ∈ S0(IRd). Dueto the symmetry relation ˜g = g, one may replace in the above statement g byg, without changing the validity of the proposition (or the assumptions).

There are various results in Gabor analysis which involve the use of theso-called adjoint lattice, which is defined in terms of commutation laws (withrespect to a given TF-lattice Λ).

Definition 4.11. For Λ � G × G we define the adjoint lattice Λ◦ asfollows:

Λ◦ = {λ◦ ∈ G× G | π(λ)π(λ◦) = π(λ◦)π(λ), ∀λ ∈ Λ}. (4.8)

Lemma 4.12. For Λ1 � G, Λ2 � G,

(Λ1 × Λ2)◦ = Λ⊥2 × Λ⊥1 , (4.9)

hence

(aZZd × bZZd)◦ =1bZZd × 1

aZZd.

The proof is an easy exercise, recalling that for H �G the orthogonal groupH⊥ � G is defined as follows:

H⊥ = {χ ∈ G , χ(h) = 1 ∀h ∈ H}.

15

It is generally true that (Λ◦)◦ = Λ, hence {π(λ◦) , λ◦ ∈ Λ◦} is a com-mutative group of TF-shifts if and only if Λ ⊆ Λ◦. For this special caseZak-transform methods can be applied. For aZZd × bZZd

� IRd × IRd

this is thecase if a · b = 1/k, for some k ∈ IN, and we speak of integer redundancy in thiscase.

One of the basic principles of Gabor analysis, called the Wexler-Raz prin-ciple [see Gro01 Thm.7.3.2, FK98 Section 7.7, FZ98 Section 3.5] helps to es-tablish a correspondence between Gabor frames and Weyl-Heisenberg familieswhich form a Riesz spaces for some closed subspace of L2(IRd).

Theorem 4.13. (Wexler-Raz Principle)The family (π(λ)g)λ∈Λ is a Gabor frame if and only if (π(λ◦)g)λ◦∈Λ◦ is a Rieszbasis. Furthermore the window g generating the dual Gabor frame (π(λ)g)λ∈Λ

coincides (up to some constant, depending only on Λ) with the window gen-erating the canonical biorthogonal Riesz basis to (π(λ◦)g)λ◦∈Λ◦ , which thus isC2

Λ · (π(λ◦)g)λ◦∈Λ◦ .As a consequence the projection PH = Pg,Λ◦ from L2(G) onto Hg,Λ◦ , the

closed linear span of the family (π(λ◦)g)λ◦∈Λ◦ within L2(IRd), can be describedby

PH(f) = C3Λ

∑λ◦∈Λ◦

〈f, gλ◦〉gλ◦ = C3Λ

∑λ◦∈Λ◦

〈f, gλ◦〉gλ◦ , f ∈ L2(G)

for some constant C3Λ depending on Λ only.

As we have seen it is of great importance for the usefulness of Weyl-Heisenberg families (gλ)λ∈Λ to know that both g and g belong to S0(IRd).Although weighted versions of this space (expressible in the terminology ofmodulation spaces, such as M1

v (IRd), cf. [Gro01], Chap. 12) are more suit-able for a quantitative description, we avoid the corresponding technicalitiesby staying with S0(IRd) for the rest of this paper. This amounts to requiringthe minimal degree of TF-concentration (much weaker than assuming that g, gbelong to S(IRd)), while still having the chance to formulate basic principles(which are not valid for general L2-windows).

Assuming that g is chosen from S0(IRd) in the first place, it is easy to checkthat g ∈ S0(IRd) if and only if the frame operator (restricted to S0(IRd)) isboundedly invertible on S0(IRd). Indeed, if this is the case then g = S−1g willbelong to S0(IRd) as well. Conversely, recall that S−1 coincides just with theframe operator induced by the dual Gabor frame, generated by (g,Λ), whichis bounded on S0(IRd) whenever g ∈ S0(IRd) (according to [FZ98]).

It is also well established that using such Gabor frames (with both g, g inS0(IRd)) one can recognize signals f ∈ S0(IRd) by the property that they have

16

absolutely summable Gabor coefficients with respect to any such system. Itis also true that the range of distributions to which such Gabor frames applyincludes S′0(IR

d).In order to avoid cumbersome terminology later on we agree to speak of a

triple (g, g,Λ) generating an S0 Gabor frame, if the situation described aboveapplies. In a similar way we will speak of tight Gabor atoms in S0(IRd) ifg ∈ S0(IRd) generates a tight Gabor frame (which then automatically has allthe good properties mentioned above).

5 Gabor Multipliers and Spline-type Spaces

The main result of this paper concerns Gabor multipliers. These are linearoperators which can be built using Gabor expansions. Similar to Fourier mul-tipliers they operate on the (canonical Gabor) coefficients through pointwisemultiplication. These operators have already been defined in Chap.3 in [FS98],where also their basic continuity properties (on L2 or S0) are discussed.

There is an essential difference between Gabor multipliers and operatorswhich are defined by multiplication on the coefficients with respect to someorthonormal system (say, Fourier multipliers). In the latter case one has avery simple symbolic calculus. Indeed, for such a case the family is closedunder composition of operators (and the product operator is defined by theproduct of the defining multipliers, and inversion can by carried out at the“symbol level”), which is not the fact for Gabor multipliers, since these arebased on a (redundant) frame.

Definition 5.14. Let g1, g2 be two L2-functions, Λ some TF-lattice forG, and (m(λ))λ∈Λ a complex-valued sequence on Λ. The Gabor multiplierassociated to the triple (g1, g2,Λ) with (strong or) upper symbol m is givenas

Gm(f) = Gg1,g2,Λ,m(f) =∑λ∈Λ

m(λ)〈f, π(λ)g1〉π(λ)g2

It is obvious from this definition (for a more precise description cf. thesubsequent paragraphs) that Gabor multipliers are essentially (infinite) linearcombinations of the rank one operators f 7→ 〈f, π(λ)g1〉π(λ)g2. Wheneverg1 = g = g2 we simple write Gg,Λ,m, then these building blocks are just theorthogonal projections onto the 1D-subspaces of L2 generated by the frameelements π(λ)g (assuming that g is normalized with ‖g‖2 = 1).

Depending on the properties of the analysis window g1, the synthesis win-dow g2 and the multiplier sequence (mλ), the overall operator Gg1,g2,Λ,m isbounded between various spaces. Typically one would assume that both g1

and g2 are Bessel-atoms with respect to the given lattice Λ, and that m is

17

bounded. In this case the coefficient mapping using g1, mapping f to thesequence of sampling values of STFTg1f over Λ maps L2(G) into `2(Λ) (bydefinition), and also the synthesis mapping c 7→

∑λ∈Λ cλπ(λ)g2 is bounded

from `2(Λ) to L2(G), and thus the overall operator defines a bounded operatoron L2(G). Following [FK98] we will assume for the rest of this article thatboth g1 and g2 belong to the Segal algebra S0(G), because then they will def-initely be Bessel atoms (for arbitrary lattices Λ). Furthermore we know thatthe overall operator is Hilbert-Schmidt in case m(λ) belongs to `2(Λ), and atrace class operator on L2(G), mapping S′0(G) into S0(G). (cf. Cor.3.3.5 in[FZ98], p. 144).

Although many of the results remain valid for the general case (g1 6= g2)we shall concentrate in the rest of this paper on the case g1 = g = g2 ∈ S0(IRd),because in this case we have the following approximate symbolic calculus:

Theorem 5.15. Assume that g ∈ S0(IRd). Then(i) Gm defines a bounded operator on (S0, L

2, S′0), and the operator norm ofGm can be estimated (up to some constant) by ‖m‖∞.(ii) The Gabor multiplier generated by m(λ) ≡ 1 is the identity operator if andonly if (g,Λ) generates a tight Gabor frame.(iii) Gm is a self-adjoint operator on L2(IRd) whenever m is real-valued.(iv) Gm is a compact operator on L2(IRd) and on S0(IRd), whenever m(λ) → 0for λ →∞ (in the sense of G);(v) m ∈ `2(Λ) implies that Gm ∈ HS, i.e., it is a Hilbert-Schmidt operator onL2(IRd), also mapping L2(IRd) into S0(IRd). Consequently the eigenvectors ofGm belong to S0(IRd) if m(λ) is real-valued.(vi) For m ∈ `1(Λ) the operator Gm is a trace class operator on L2(IRd),mapping S′0(IR

d) into S0(IRd).Proof. These statements follow more or less directly from the boundedness

properties of the coefficient resp. synthesis mappings (for fixed lattice Λ), asdescribed in some detail in Section 3.3.3 in [FZ98].

The findings of the above Theorem can be reformulated now for generalpairs (g1, g2), using the terminology of Gelfand triples as follows:

Theorem 5.16. For every pair (g1, g2) in S0(IRd), and any TF-latticeΛ, the mapping from the strong symbol (multiplier) (m(λ))λ∈Λ to the corre-sponding Gabor multiplier maps the Gelfand triple

(`1(Λ), `2(Λ), `∞(Λ)

)into

the bounded linear operators with kernel in the corresponding Gelfand triple(S0(IRd × IR

d), L2(IRd × IR

d), S′0(IR

d × IRd)).

Once we have established these basic results on Gabor multipliers weare able to turn to more interesting questions concerning the mapping(m(λ))λ∈Λ 7→ Gm. In particular we are interested to find out under which

18

conditions this mapping is injective (not only on `2(Λ), but also on `∞(Λ)),and how—in the positive case—the symbol can be recovered from the operator(in other words we want to establish the bounded invertibility of this mapping).

Remark: That there is a good chance that the operator family (Pλ)λ∈Λ isa Riesz basis for its closed linear span (despite the fact that the family (gλ)λ∈Λ

is “redundant”, hence truly linearly dependent in the case of Gabor analysisover ZN ) has been verified numerically some time ago, cf. [FH01]. Althoughthis is perhaps a surprising result to some readers (as it was to ourselves) weonly want to mention two facts: While linear independence of a family (gλ)λ∈Λ

certainly implies that (Pλ)λ∈Λ is a linear independent family, the converseis apparently not true. After some thought this should not be surprising.Assuming that Λ is a TF-lattice (for ZN ) of redundancy red = N/(a ∗ b) weare only speaking of red ∗N independent operators in the N ∗ red-dimensionalspace of all matrices, while of course (for red > 1) N ∗ red Gabor atoms haveto be linear dependent in the N -dimensional CN of signals over ZN .

Our main result concerning the mapping m 7→ Gm follows next:Theorem 5.17. Assume that (g,Λ) generates an S0-Gabor frame for

L2(IRd), with ‖g‖2 = 1, and write P for the projection f 7→ 〈f, g〉g, andPλ := ρ(λ)P . Then the family (Pλ)λ∈Λ is a Riesz basis in the Hilbert spaceHS of all Hilbert-Schmidt operators on L2(IRd) if and only if the functionH(s), obtained by taking the Λ-Fourier transform of

(|STFTg(g)(λ)|2

)λ∈Λ

, isfree of zeros. The canonical biorthogonal family is of the form (Qλ)λ∈Λ, withQλ = ρ(λ)Q for all λ ∈ Λ and some operator Q with kernel in S0(IR2d). Inparticular, the best approximation of a given Hilbert-Schmidt operator T byGabor multipliers (for fixed pair (g,Λ)) is of the form

PG(T ) :=∑λ∈Λ

〈T, ρ(λ)Q〉HS ρ(λ)P. (5.10)

The mapping PG is an orthogonal projection within the class HS.

Corollary 5.18. The restriction of the mapping PG defined above to theclass B of all operators with kernels in S0(IR2d) is continuous on that class,while on the other hand PG is well defined and bounded on L(S0, S

′0). In

particular, PG(T ) is a well defined operator from S0(IRd) to S′0(IRd) for every

bounded linear operator T on L2(IRd).Proof of Thm. 5.17: The result relies on the fact that the entries of the

Gram matrix for the system (Pλ)λ∈Λ are the Hilbert-Schmidt scalar products

(〈ρ(λ)P, ρ(λ′)P 〉)λ,λ′∈Λ .

19

Since P = g ⊗ g∗, a simple reformulation yields for these entries

〈Tλσ(P ), Tλ′σ(P )〉HS = 〈ρ(λ)P, ρ(λ′)P 〉HS = |〈π(λ)g, π(λ′)〉|2 =

= |〈g, π(λ′ − λ)g〉|2 = |STFTgg(λ′ − λ)|2,

i.e., the Gram matrix is circulant (with respect to the index set Λ, understoodas discrete Abelian group). It is clear that this matrix is boundedly invertibleif and only if the transfer function corresponding to the convolution kerneldefined by

(|STFTg(g)(λ)|2

)λ∈Λ

is bounded away from zero, as stated in thetheorem.

Thus we only have to show that Q ∈ B. As it is easy to verify thatκ(P ) ∈ S0(IR2d) if (and only if) g ∈ S0(IRd), we only have to recall that thebijection between the kernels of operators and corresponding Kohn-Nirenbergsymbols is not only unitary (at the Hilbert space level), but also preservesmembership in S0. It now follows that Thm. 3.6 applies (if and only if thenon-vanishing condition stated above is satisfied). The rest of the main resultis then an immediate consequence of general facts about operators with kernelsin S0(IR2d), L2(IR2d), and S′0(IR

2d) respectively.The connection to spline theory will be based on the use of the so-called

Kohn-Nirenberg symbol σ(T ) associated with each operator T ∈ L(S0, S′0). For

operators T which have their kernel κ(T ) in S0(IR2d), the symbol σ(T ) is thecontinuous function defined over the TF-plane G × G through the followingformula (cf. [FK98], Thm.7.5.1):

σ(T )(x, ξ) =∫

G

κ(T )(x, x− y)e−2πiξydy (5.11)

This mapping from kernels to KN-symbols extends to all of L(S0, S′0).

Indeed, using the invariance properties of S0(IRd) under automorphisms ofIRd and partial Fourier transforms it is not difficult to see that the mappingκ(T ) 7→ σ(T ) is well-defined as a linear and invertible mapping from S0(IR2d) to

S0(IRd × IRd), which extends to a unitary mapping between Hilbert-Schmidt

Operators and KN-symbols in L2(IRd × IRd) (the traditional point of view).

This mapping, restricted to B, establishes an isomorphism onto S0(IRd × IRd).

(cf. [FK98] Thm.7.5.1 for details).The following lemma (cf. [FK98], Lemma 7.5.3) contains the key to the

identification of spaces of Gabor multipliers with spline-type spaces on IR2d.Lemma 5.19. (Shift Covariance of the KN-symbol)

The Kohn-Nirenberg mapping T 7→ σ(T ) intertwines the action of Λ�IRd × IRd

20

on the Hilbert-Schmidt operators, given by T 7→ ρ(λ)T , and the regular repre-

sentation of Λ � IRd × IRd

on L2(IRd × IRd); in other words, one has for any

T ∈ L(S0, S′0) and λ1 = (x1, ξ1):

σ[ρ(λ1)(T )] = σ[π∗(λ1) ◦ T ◦ π(λ1)] = Tλ1 [σ(T )] (5.12)

This lemma is the key to establishing the bridge between the principalplayers of this paper, i.e., Gabor multipliers and spline-type spaces.

Theorem 5.20. (Gabor Multipliers and Spline-type Spaces)Assume that g1, g2 are in S0(IRd), and that Λ is a TF-lattice for IRd, i.e., of theform Λ = A(ZZ2d) for some non-singular matrix 2d× 2d-matrix A. Then T isa Gabor multiplier of the form {Gg1,g2,m,Λ | m(λ) ∈ `2(Λ)}, for a suitablesequence (m(λ))λ∈Λ, if and only if σ(T ) belongs to the spline-type space over

IRd × IRd

which is generated by the S0(IRd × IRd)-function

[σ(g1 ⊗ g∗2)](x, ξ) = g1(x)g2(ξ)e−2πix·ξ. (5.13)

In particular, the space of Gabor multipliers with `2(Λ)-symbols, i.e.

{Gg1,g2,m,Λ | m(λ) ∈ `2(Λ)}

is a closed subspace of HS if the functions (Tλσ(g1⊗ g∗2))λ∈Λ are a Riesz basisfor their closed linear span.

Proof. The proof of this statement is a simple combination of the state-ments given above and a direct computation of the Kohn-Nirenberg symbolfor a rank-one operator (cf. [FK98], p.248). Indeed, the mapping T 7→ σ(T )restricted to the class of Gabor-multipliers of the given form (i.e., using (g,Λ))establishes an isomorphism between the corresponding spaces. The mappingcan first be viewed at the L2-level (from HS to L2(IRd × IR

d)), but of course

it is valid for the other levels as well.Definition 5.21. For any linear operator T ∈ HS we define the lower

symbol with respect to (g,Λ) as the function over Λ given by

lT (λ) = 〈T, Pλ〉HS = 〈T (π(λ)g), π(λ)g〉, λ ∈ Λ. (5.14)

Remark: Due to the fact that the scalar product can be expressed by aduality pairing between their kernels, i.e., 〈T, S〉HS = 〈κ(T ), κ(S)〉, and thefact that Pλ ∈ B for g1, g2 ∈ S0(IRd), it follows that for g ∈ S0(IRd) the abovedefinition extends to L(S0, S

′0) in a natural way.

Proposition 5.22. Assume that (Pλ)λ∈Λ with g ∈ S0(IRd) forms a Rieszbasis within HS. Then for any Gabor multiplier Gm with m(λ) ∈ `∞(Λ), the

21

recovery from its lower symbol is possible, according to the formula:

Gm =∑λ∈Λ

〈T (π(λ)g), π(λ)g〉HS Qλ =∑λ∈Λ

lT (λ) Qλ. (5.15)

Moreover, (m(λ)) ∈ `2(Λ) if and only if lT ∈ `2(Λ), and (m(λ)) ∈ `1(Λ) ifand only if lT ∈ `1(Λ) (and the series converges in HS or B resp.).

Proposition 5.23. The orthogonal projection from the space of allHilbert-Schmidt operators onto the Gabor multipliers (for fixed (g,Λ)) is notonly continuous in the sense of the HS-norm. Its restriction to B is continuouson B, and also extends to the larger space L(S0, S

′0), yielding Gabor multipliers

with strong symbol in `∞(Λ).

AcknowledgementsThe author would like to thank Werner Kozek for many discussions on thequantization of operators and for convincing me that the Kohn-Nirenberg sym-bol is of utmost importance for the understanding of the properties of operatorsarising in signal processing applications.

References

[AG01] A. Aldroubi, K. Grochenig: Non-uniform sampling in spline-typespaces, SIAM Rev., 2001, to appear.

[AST01] A. Aldroubi, Q. Sun, W.S. Tang: p-Frames and shift invariantsubspaces of Lp, J. Four. Anal. Appl. 7/1 (2001), 2–21.

[BL98] J. J. Benedetto, S. Li: The theory of multiresolution analysis framesand applications to filter banks, Appl. Comput. Harmon. Anal., 5 (1998), 389–427.

[Fei80] H.G. Feichtinger: Un espace de distributions temperees sur lesgroupes, Compt. Rend. Acad. Sci. Paris, Ser.A, 290 (17) (1980), 791–794.

[FG-I] H. G. Feichtinger, K. Grochenig: Banach spaces related to integrablegroup representations and their atomic decompositions, I. J. Funct. Anal., 861989, 307–340.

[FG-II] H. G. Feichtinger, K. Grochenig: Banach spaces related to inte-grable group representations and their atomic decompositions, I. Monatsh. f.Math., 108 1989, (129–148).

[FG92] H. G. Feichtinger, K. Grochenig: Gabor wavelets and the Heisenberggroup: Gabor expansions and Short Time Fourier transform from the grouptheoretical point of view. In C.K. Chui, editor, Wavelets – A Tutorial in Theoryand Applications. Academic Press, Boston, 1992, 359–397.

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[FG97] H. G. Feichtinger, K. Grochenig: Gabor frames and time-frequencyanalysis of distributions J. Funct. Anal. 146/2 (1997), 464-495.

[FH01] H. G. Feichtinger, M. Hampejs: Best approximation of linear time-variant systems by Gabor multipliers, preprint, 2001.

[FK98] H. G. Feichtinger, W. Kozek: Quantization of TF lattice-invariantoperators on elementary LCA groups, Chap. 7 in [FS98].

[FS98] H. G. Feichtinger, T. Strohmer (Eds.): Gabor Analysis and Algo-rithms, Theory and Applications. Birkhauser, Boston, 1998.

[FZ98] H. G. Feichtinger, G. Zimmermann: A Banach space of test func-tions for Gabor analysis, Chap. 3 in [FS98].

[Fei02] H. G. Feichtinger: A First Survey of Gabor Multipliers, Chapter in[FS02], to appear 2002.

[FS02] H. G. Feichtinger, T. Strohmer (Eds.): Advances in Gabor Analysis.Birkhauser, Boston, 2002.

[Fol89] G. B. Folland: Harmonic Analysis in Phase Space, Annals of Math.Studies. Princeton Univ. Press (NJ), 1989.

[FJ1] M. Frazier, B. Jawerth: Decomposition of Besov spaces. IndianaUniv. Math. J. 34 (1985), 777–799.

[FJ2] M. Frazier, B. Jawerth: The φ-transform and applications to distri-bution spaces. Proc. Conf. Lund 1986, “Function Spaces and Applications”,Lecture Notes in Math. 1302 (1988), 223–246.

[Gro01] K. Grochenig: Foundations of Time-Frequency Analysis.Birkhauser, Boston, 2001.

[GL01] K. Grochenig, M. Leinert: Wiener’s Lemma for twisted convolutionand Gabor frames, 2001, submitted.

[Jia97] R.-Q. Jia: Shift-invariant spaces on the real line, Proc. Amer.Math. Soc. 125 (1997).

[OCL89] J. P. Oakley, M. J. Cunningham, G. Little: A Fourier-domainformula for the least-squares projection of a function onto a repetitive basis inN-dimensional space, IEEE Trans. ASSP, Vol. 38/1 (1990), 114–120.

[RS01] H. Reiter, I. Stegeman: Classical Harmonic Analysis and LocallyCompact Groups, 2nd Ed., Oxford Univ. Press, 2001.

[You80] R. Young: An Introduction to Nonharmonic Fourier. Acad.Press, New York, 1980.

[GZDiss] G. Zimmermann: Projective Multiresolution Analysis and Gen-eralized Sampling, PhD Thesis, Univ. Maryland, College Park, 1994.

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